March  2020, 19(3): 1421-1448. doi: 10.3934/cpaa.2020052

Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data

1. 

School of Mathematics, South China University of Technology, Guangzhou 510641, China

2. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

* Corresponding author

Received  April 2019 Revised  June 2019 Published  November 2019

In this paper, we study the asymptotic behavior of global spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations for the viscous heat conducting ideal polytropic gas flow with large initial data in $ H^1 $, when the heat conductivity coefficient depends on the temperature, practically, $ \kappa(\theta) = \tilde{\kappa}_1+\tilde{\kappa}_2\theta^q $ where constants $ \tilde{\kappa}_1>0 $, $ \tilde{\kappa}_2>0 $ and $ q>0 $ (as to the case of $ \tilde{\kappa}_1 = 0 $, please refer to the Appendix). In addition, the exponential decay rate of solutions toward to the constant state as time tends to infinity for the initial boundary value problem in bounded domain is obtained. The mass density and temperature are proved to be pointwise bounded from below and above, independent of time although strong nonlinearity in heat diffusion. The analysis is based on some delicate uniform energy estimates independent of time.

Citation: Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
References:
[1] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.   Google Scholar
[2]

H. B. Cui and Z. A. Yao, Asympyopic behavior of compressible p-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.  Google Scholar

[3]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[4]

R. DuanA. Guo and C. J. Zhu, Global strong solution to compressible Navier-Stokes equations with density dependent viscosity and temperature dependent heat conductivity, J. Differential Equations, 262 (2017), 4314-4335.  doi: 10.1016/j.jde.2017.01.007.  Google Scholar

[5]

H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156.  doi: 10.1137/S003614109834394X.  Google Scholar

[6]

D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5.  Google Scholar

[7]

L. Hsiao and T. Luo, Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1277-1296.  doi: 10.1017/S0308210500023404.  Google Scholar

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.  Google Scholar

[9]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.  Google Scholar

[10]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl., 175 (1998), 253-275.  doi: 10.1007/BF01783686.  Google Scholar

[11]

S. Jiang, Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190 (1998), 169-183.  doi: 10.1002/mana.19981900109.  Google Scholar

[12]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar

[13]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X.  Google Scholar

[14]

B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2$^{nd}$ edition, Pergamon Press Ltd., OXford, 1987.  Google Scholar

[17]

Z. L. Li and Z. H. Guo, On free boundary problem for compressible Navier-Stokes equations with temperature-dependent heat conductivity, Discrete Contin.Dyn.Syst.-B, 22 (2017), 3903-3919.  doi: 10.3934/dcdsb.2017201.  Google Scholar

[18]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar

[19]

H.X. LiuT. YangH. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.  Google Scholar

[20]

T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Appl. Math., 5 (1988), 205-224.  doi: 10.1007/BF03167873.  Google Scholar

[21]

V. B. Nikolaev, Solvability of a mixed problem for equations of one-dimensional axisymmetric motion of a viscous gas, Dinamika Sploshn. Sredy, 175 (1980), 83-92.   Google Scholar

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[23]

X. L. Qin and Z. A. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 244 (2008), 2041-2061.  doi: 10.1016/j.jde.2007.11.001.  Google Scholar

[24]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal., 216 (2015), 1049-1086.  doi: 10.1007/s00205-014-0826-x.  Google Scholar

[25]

Y. M. Qin, Exponential stability for the compressible Navier-Stokes equations with the cylinder symmetry in $R^3$, Nonlinear Anal. Real World Appl., 11 (2010), 3590-3607.  doi: 10.1016/j.nonrwa.2010.01.006.  Google Scholar

[26]

Y. M. Qin and L. M. Jiang, Global existence and exponential stability of solutions in $H^4$ for the compressible Navier-Stokes equations with the cylinder symmetry, J. Differential Equations, 249 (2010), 1353-1384.  doi: 10.1016/j.jde.2010.05.019.  Google Scholar

[27]

L. Wan and T. Wang, Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data, Nonlinear Anal. Real World Appl., 39 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.06.006.  Google Scholar

[28]

L. Wan and T. Wang, Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients, J. Differential Equations, 262 (2017), 5939-5977.  doi: 10.1016/j.jde.2017.02.022.  Google Scholar

[29]

T. Wang and H. J. Zhao, One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 26 (2016), 2237-2275.  doi: 10.1142/S0218202516500524.  Google Scholar

[30] Y. B. Zekdovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, vol. II, Academic Press, New York, 1967.   Google Scholar

show all references

References:
[1] S. Chapman and T. G. Cowling, The Mathematical Theory of Nonuniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press, Cambridge, 1990.   Google Scholar
[2]

H. B. Cui and Z. A. Yao, Asympyopic behavior of compressible p-th power Newtonian fluid with large initial data, J. Differential Equations, 258 (2015), 919-953.  doi: 10.1016/j.jde.2014.10.011.  Google Scholar

[3]

C. M. Dafermos, Global smooth solutions to the initial-boundary value problem for the equations of one-dimensional nonlinear thermoviscoelasticity, SIAM J. Math. Anal., 13 (1982), 397-408.  doi: 10.1137/0513029.  Google Scholar

[4]

R. DuanA. Guo and C. J. Zhu, Global strong solution to compressible Navier-Stokes equations with density dependent viscosity and temperature dependent heat conductivity, J. Differential Equations, 262 (2017), 4314-4335.  doi: 10.1016/j.jde.2017.01.007.  Google Scholar

[5]

H. Frid and V. Shelukhin, Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry, SIAM J. Math. Anal., 31 (2000), 1144-1156.  doi: 10.1137/S003614109834394X.  Google Scholar

[6]

D. Hoff and H. K. Jenssen, Symmetric nonbarotropic flows with large data and forces, Arch. Ration. Mech. Anal., 173 (2004), 297-343.  doi: 10.1007/s00205-004-0318-5.  Google Scholar

[7]

L. Hsiao and T. Luo, Large-time behaviour of solutions for the outer pressure problem of a viscous heat-conductive one-dimensional real gas, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 1277-1296.  doi: 10.1017/S0308210500023404.  Google Scholar

[8]

H. K. Jenssen and T. K. Karper, One-dimensional compressible flow with temperature dependent transport coefficients, SIAM J. Math. Anal., 42 (2010), 904-930.  doi: 10.1137/090763135.  Google Scholar

[9]

S. Jiang, On the asymptotic behavior of the motion of a viscous, heat-conducting, one-dimensional real gas, Math. Z., 216 (1994), 317-336.  doi: 10.1007/BF02572324.  Google Scholar

[10]

S. Jiang, Large-time behavior of solutions to the equations of a viscous polytropic ideal gas, Ann. Mat. Pura Appl., 175 (1998), 253-275.  doi: 10.1007/BF01783686.  Google Scholar

[11]

S. Jiang, Global smooth solutions of the equations of a viscous, heat-conducting, one-dimensional gas with density-dependent viscosity, Math. Nachr., 190 (1998), 169-183.  doi: 10.1002/mana.19981900109.  Google Scholar

[12]

S. Jiang, Large-time behavior of solutions to the equations of a one-dimensional viscous polytropic ideal gas in unbounded domains, Comm. Math. Phys., 200 (1999), 181-193.  doi: 10.1007/s002200050526.  Google Scholar

[13]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268.  doi: 10.1137/07070005X.  Google Scholar

[14]

B. Kawohl, Global existence of large solutions to initial-boundary value problems for a viscous, heat-conducting, one-dimensional real gas, J. Differential Equations, 58 (1985), 76-103.  doi: 10.1016/0022-0396(85)90023-3.  Google Scholar

[15]

A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9.  Google Scholar

[16]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2$^{nd}$ edition, Pergamon Press Ltd., OXford, 1987.  Google Scholar

[17]

Z. L. Li and Z. H. Guo, On free boundary problem for compressible Navier-Stokes equations with temperature-dependent heat conductivity, Discrete Contin.Dyn.Syst.-B, 22 (2017), 3903-3919.  doi: 10.3934/dcdsb.2017201.  Google Scholar

[18]

J. Li and Z. L. Liang, Some uniform estimates and large-time behavior of solutions to one-dimensional compressible Navier-Stokes system in unbounded domains with large data, Arch. Ration. Mech. Anal., 220 (2016), 1195-1208.  doi: 10.1007/s00205-015-0952-0.  Google Scholar

[19]

H.X. LiuT. YangH. J. Zhao and Q. Y. Zou, One-dimensional compressible Navier-Stokes equations with temperature dependent transport coefficients and large data, SIAM J. Math. Anal., 46 (2014), 2185-2228.  doi: 10.1137/130920617.  Google Scholar

[20]

T. Nagasawa, Global asymptotics of the outer pressure problem with free boundary, Japan J. Appl. Math., 5 (1988), 205-224.  doi: 10.1007/BF03167873.  Google Scholar

[21]

V. B. Nikolaev, Solvability of a mixed problem for equations of one-dimensional axisymmetric motion of a viscous gas, Dinamika Sploshn. Sredy, 175 (1980), 83-92.   Google Scholar

[22]

R. H. Pan and W. Z. Zhang, Compressible Navier-Stokes equations with temperature dependent heat conductivity, Commun. Math. Sci., 13 (2015), 401-425.  doi: 10.4310/CMS.2015.v13.n2.a7.  Google Scholar

[23]

X. L. Qin and Z. A. Yao, Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity, J. Differential Equations, 244 (2008), 2041-2061.  doi: 10.1016/j.jde.2007.11.001.  Google Scholar

[24]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal., 216 (2015), 1049-1086.  doi: 10.1007/s00205-014-0826-x.  Google Scholar

[25]

Y. M. Qin, Exponential stability for the compressible Navier-Stokes equations with the cylinder symmetry in $R^3$, Nonlinear Anal. Real World Appl., 11 (2010), 3590-3607.  doi: 10.1016/j.nonrwa.2010.01.006.  Google Scholar

[26]

Y. M. Qin and L. M. Jiang, Global existence and exponential stability of solutions in $H^4$ for the compressible Navier-Stokes equations with the cylinder symmetry, J. Differential Equations, 249 (2010), 1353-1384.  doi: 10.1016/j.jde.2010.05.019.  Google Scholar

[27]

L. Wan and T. Wang, Asymptotic behavior for cylindrically symmetric nonbarotropic flows in exterior domains with large data, Nonlinear Anal. Real World Appl., 39 (2018), 93-119.  doi: 10.1016/j.nonrwa.2017.06.006.  Google Scholar

[28]

L. Wan and T. Wang, Symmetric flows for compressible heat-conducting fluids with temperature dependent viscosity coefficients, J. Differential Equations, 262 (2017), 5939-5977.  doi: 10.1016/j.jde.2017.02.022.  Google Scholar

[29]

T. Wang and H. J. Zhao, One-dimensional compressible heat-conducting gas with temperature-dependent viscosity, Math. Models Methods Appl. Sci., 26 (2016), 2237-2275.  doi: 10.1142/S0218202516500524.  Google Scholar

[30] Y. B. Zekdovich and Y. P. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, vol. II, Academic Press, New York, 1967.   Google Scholar
[1]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263

[2]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[3]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57

[4]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[5]

Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008

[6]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[7]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675

[8]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

[9]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[10]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609

[11]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719

[12]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

[13]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052

[14]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055

[15]

Takeshi Taniguchi. The exponential behavior of Navier-Stokes equations with time delay external force. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 997-1018. doi: 10.3934/dcds.2005.12.997

[16]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[17]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010

[18]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079

[19]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073

[20]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (21)
  • HTML views (33)
  • Cited by (0)

Other articles
by authors

[Back to Top]